The free energy in a class of quantum spin systems and interchange processes

TL;DR

This paper analyzes quantum spin systems on complete graphs, deriving free energy and critical temperature, and reveals phase transition behaviors, connecting quantum models with classical Potts models.

Contribution

It provides a unified analysis of quantum spin systems for various spins, linking their phase transitions to classical Potts models and determining free energy explicitly.

Findings

Critical temperature matches that of the classical Potts model.

Phase transition is discontinuous for spins S ≥ 1.

Results recover known cases for S=1/2.

Abstract

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin $S=\tfrac12$ the model is the Heisenberg ferromagnet, for general spin $S\in\tfrac12\mathbb{N}$ it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by T\'oth and by Penrose when $S=\tfrac12$). The critical temperature is shown to coincide (as a function of $S$) with that of the $q=2S+1$ state classical Potts model, and the phase transition is discontinuous when $S\geq1$.